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Real Analysis, differentiation

  1. May 8, 2008 #1
    Solved: Real Analysis, differentiation

    1. The problem statement, all variables and given/known data
    If g is differentiable and g(x+y)=g(x)(g(y) find g(0) and show g'(x)=g'(0)g(x)

    3. The attempt at a solution
    I solved g(0)=1

    and

    I got as far as

    [tex]
    g'(x)=\lim_{\substack{x\rightarrow 0}}g(x) \frac{g(h)-1}{h}
    [/tex]

    but now I am stuck.

    Thank you in advance
     
    Last edited: May 8, 2008
  2. jcsd
  3. May 8, 2008 #2
    Never mind... Just did g'(0) and plug in...
     
  4. May 8, 2008 #3

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    Your formula for g' is incorrect- though it may be a typo.
    [tex]g'(x)= \lim_{\substack{h\rightarrow 0}}g(x)\frac{g(h)- 1}{h}[/tex]
    where the limit is taken as h goes to 0, not x. Since "g(x)" does not depend on h, you can factor that out:
    [tex]g'(x)= g(x)\left(\lim_{\substack{h\rightarrow 0}}\frac{g(h)-1}{h}\right)[/tex]
    and you should be able to see that the limit is simply the definition of g'(0).
     
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